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arxiv: 2511.16776 · v3 · pith:KLQGPIEGnew · submitted 2025-11-20 · ⚛️ physics.flu-dyn

On the Poisson-Source Basis of Logarithmic Wall-Pressure-Variance Growth

Jonathan M. O. Massey , Joseph C. Klewicki , Beverley J. McKeon This is my paper

Pith reviewed 2026-05-21 18:25 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords wall-pressure variancepressure Poisson equationlogarithmic scalingwall-bounded turbulenceReynolds numberdirect numerical simulationuniform momentum zonesvortical fissures
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The pith

The nonlinear source in the pressure-Poisson equation supplies the logarithmic coefficient of inner-scaled wall-pressure variance while the linear source supplies a Reynolds-number-independent offset.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a mechanistic link between the two source terms in the incompressible pressure-Poisson equation and the observed logarithmic growth of inner-scaled wall-pressure variance with frictional Reynolds number. It separates a linear source tied to mean shear from a nonlinear source built from quadratic velocity fluctuations, then traces each through the integral solution for wall pressure. A sympathetic reader would care because the separation explains why the variance grows as ln δ⁺ rather than some other power or constant, connecting an empirical scaling to specific turbulence structures. The linear term is shown to reside mainly in the buffer layer and to become invariant under inner scaling, while the nonlinear term concentrates in the fissures between uniform momentum zones and grows with the extent of the inertial layer. This supplies a concrete carrier for the Reynolds-number dependence without invoking additional modeling assumptions.

Core claim

To leading order the linear source provides a Reynolds-number-independent offset, while the nonlinear source contributes the logarithmic coefficient in the inner-scaled wall-pressure variance representation. The linear source contribution sits predominantly in the buffer layer and maps to the near-wall cycle, becoming δ⁺ invariant under inner scaling. The interfacial regions between uniform momentum zones, called vortical fissures, spatially localise strain and vorticity and contain an increasing fraction of the nonlinear source, thereby linking the changing nonlinear contribution directly to the ln δ⁺ growth.

What carries the argument

Decomposition of the incompressible pressure-Poisson equation into linear (rapid, mean-shear) and nonlinear (slow, quadratic-fluctuation) source terms, followed by their separate integration to obtain wall-pressure variance contributions under inner scaling.

If this is right

  • The linear source contribution is δ⁺ invariant under inner scaling and therefore adds only an offset to the logarithmic representation.
  • Fissures act as compact carriers for the source terms, with the nonlinear term especially prominent in these regions.
  • The increasing proportion of strain and vorticity contained in fissures accounts for the growth of the nonlinear contribution with inertial-layer extent.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the separation holds at higher Reynolds numbers it would allow targeted modeling of the log coefficient through inertial-layer statistics alone.
  • Resolving the fissure regions may be sufficient to capture the Reynolds-number dependence of pressure variance in large-eddy simulations.
  • The same source decomposition could be applied to other near-wall statistics such as wall-shear-stress variance to test for analogous linear-nonlinear splits.

Load-bearing premise

That the leading-order separation of linear and nonlinear source contributions remains valid when extrapolated from the single DNS at δ⁺ ≈ 550 to the high-Reynolds-number limit where the logarithmic growth is observed.

What would settle it

A DNS at frictional Reynolds number several times larger than 550 that extracts the separate linear and nonlinear source contributions to wall-pressure variance and checks whether the nonlinear part continues to grow as ln δ⁺ while the linear part remains flat.

Figures

Figures reproduced from arXiv: 2511.16776 by Beverley J. McKeon, Jonathan M. O. Massey, Joseph C. Klewicki.

Figure 1
Figure 1. Figure 1: (a) Schematic of the wall-pressure spectra from [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Instantaneous and mean r.m.s. profiles of the pressure across a half-channel [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Ring-averaged 𝑦 +–𝑘 spectra of (a) linear L and (b) nonlinear Q sources, as well as the corresponding wall-pressure attribution maps to the bottom wall for (c) linear pressure and (d) nonlinear pressure. (c,d) The black dashed line indicates 𝑘 𝑦 = 1, and the two gray lines indicate 𝜉 = 0.3 and 𝜉 = 3. The kink in the lines is due to the change from log to linear scaling at 𝑦 + = 10 toward the channel wall. … view at source ↗
Figure 4
Figure 4. Figure 4: Profiles of the r.m.s. of the Poisson source components versus [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) Wall-normal trends at 𝛿 + ≈ 550: r.m.s.(𝜕𝑥 𝑣) (blue), |2Ω𝑧 | (green), and r.m.s.(L) (black). Shaded bands mark the buffer and logarithmic windows. (b) Premultiplied, inner-scaled wall-pressure spectrum carried by L, shown as 𝑘 +𝐸 + 𝑝L (𝑘 + ), when the integral in equation (2.7) is restricted to three slabs on the lower half-channel: 5 ⩽ 𝑦 + < 30 (blue), 30 ⩽ 𝑦 + < 2.6 √ 𝛿 + (orange; ∼60 at 𝛿 + ≈550), a… view at source ↗
Figure 6
Figure 6. Figure 6: Schematic of the physical process that yields the ln [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
read the original abstract

In high-Reynolds-number wall-bounded flows, the inner-scaled wall-pressure variance \ra{is often represented as a} logarithmic increase with frictional Reynolds number. We consider the two sources of the incompressible pressure--Poisson equation: a linear (rapid) term linked to mean shear and a nonlinear (slow) term composed of quadratic velocity fluctuations. This paper establishes a link between the sources and the coefficients in \rtwo{a logarithmic} inner-scaled variance \rtwo{representation}. \rone{To leading order} we \rone{posit that} the \ra{linear source provides a Reynolds-number-independent offset, while the nonlinear source contributes the logarithmic coefficient}. The illustrative dataset is direct numerical simulation (DNS) at \rtwo{frictional Reynolds number} $\delta^+\approx 550$, although the principal contribution is the establishment of a mechanistic link to well-known high-$\delta^+$ scalings of wall-bounded turbulence. Through consideration of the sources and the integral solution method of the Poisson equation, we find that the linear source contribution sits predominantly in the buffer layer and maps to the near-wall cycle. \rone{To leading order}, this contribution becomes $\delta^+$ invariant under inner scaling, thus contributing an offset in the \rtwo{logarithmic representation}. The interfacial regions between uniform momentum zones characteristic of the inertial layer (vortical fissures) spatially localise strain and vorticity contributions and contain an increasingly large proportion of the strain and vorticity. We show that fissures act as a compact carrier for the source terms, with the nonlinear term especially prominent in these regions. By considering the inertial layer statistics, we link the changing nonlinear contribution to \rtwo{the} $\ln \delta^+$ \rtwo{growth}, in agreement with previous empirical observations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript posits that the logarithmic growth of inner-scaled wall-pressure variance with frictional Reynolds number δ⁺ in high-Re wall-bounded flows originates from the sources in the incompressible pressure-Poisson equation. To leading order, the linear (rapid) source tied to mean shear supplies a δ⁺-independent offset localized in the buffer layer and near-wall cycle, while the nonlinear (slow) source, concentrated in the vortical fissures between uniform momentum zones in the inertial layer, supplies the logarithmic coefficient as the inertial layer thickens and fissures occupy a larger fraction. The argument is developed from DNS at a single Reynolds number δ⁺≈550 together with the integral solution of the Poisson equation.

Significance. If the leading-order linear/nonlinear separation and its Re-independence hold upon extrapolation, the work supplies a mechanistic, parameter-free account of the observed logarithmic scaling that directly ties pressure statistics to established structural features of wall turbulence (near-wall cycle, uniform momentum zones, and fissures). It avoids fitted parameters and offers a concrete physical carrier for the nonlinear contribution without contradicting known scalings.

major comments (1)
  1. [Abstract] Abstract: the leading-order posit that the linear source is δ⁺-invariant while the nonlinear source supplies the logarithmic coefficient is demonstrated only on DNS at δ⁺≈550. No quantitative test or scaling argument is given to show that the relative weighting of the two sources (or the fissure statistics) remains unchanged when extrapolated to the high-Re regime in which the logarithmic representation is observed; this assumption is load-bearing for the claimed link between source decomposition and ln δ⁺ growth.
minor comments (2)
  1. The abstract contains several LaTeX editing artifacts (e.g., ra, rtwo, rone) that should be removed for readability.
  2. A brief statement of the precise assumptions required for the leading-order source separation (e.g., neglect of higher-order interactions) would improve clarity in the discussion of the integral solution method.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive review and for highlighting the potential significance of linking Poisson sources to the observed logarithmic scaling. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the leading-order posit that the linear source is δ⁺-invariant while the nonlinear source supplies the logarithmic coefficient is demonstrated only on DNS at δ⁺≈550. No quantitative test or scaling argument is given to show that the relative weighting of the two sources (or the fissure statistics) remains unchanged when extrapolated to the high-Re regime in which the logarithmic representation is observed; this assumption is load-bearing for the claimed link between source decomposition and ln δ⁺ growth.

    Authors: We agree that the analysis relies on DNS at a single Reynolds number (δ⁺≈550) and does not include a direct quantitative verification of source weighting or fissure volume fraction at higher Re. This is a genuine limitation of the present study. The manuscript's central contribution is nevertheless the identification of a leading-order mechanistic connection: the linear source maps to the near-wall cycle (whose inner-scaled statistics are Re-independent by definition), while the nonlinear source is localized to fissures whose relative volume grows with inertial-layer thickness. The latter supplies the scaling argument for the logarithmic coefficient, drawing on established structural features (uniform momentum zones and fissures) that are documented in the literature to persist and scale with δ⁺. We will revise the abstract and add a clarifying paragraph in the discussion to state these assumptions explicitly and to emphasize that the link is mechanistic rather than a fitted extrapolation from the single dataset. revision: partial

Circularity Check

0 steps flagged

No significant circularity; mechanistic link derived from DNS source decomposition without reduction to fitted inputs or self-citations

full rationale

The paper posits a leading-order separation in which the linear source supplies a Reynolds-number-independent offset and the nonlinear source supplies the logarithmic coefficient in the inner-scaled wall-pressure variance. This separation is demonstrated directly via the Poisson integral solution and source decomposition applied to the single DNS at δ⁺ ≈ 550, localizing linear contributions to the buffer layer (hence inner-scaled invariant) and nonlinear contributions to inertial-layer fissures whose increasing proportion is then linked to ln δ⁺ growth. The logarithmic growth itself is referenced to prior empirical observations rather than fitted or defined within the present dataset, and no self-citation chain or ansatz is invoked to force the result. The derivation therefore remains self-contained against external benchmarks and does not reduce any claimed prediction to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard decomposition of the incompressible pressure Poisson equation into linear and nonlinear terms, the validity of inner scaling for the linear contribution, and the representativeness of a single moderate-Re DNS for asymptotic high-Re behavior. No new free parameters or invented entities are introduced.

axioms (2)
  • standard math The incompressible Navier-Stokes equations hold and the pressure satisfies the Poisson equation obtained by taking the divergence.
    Invoked in the opening paragraph when the two sources are identified.
  • domain assumption Inner scaling renders the linear source contribution Reynolds-number independent to leading order.
    Stated explicitly as the leading-order posit for the offset term.

pith-pipeline@v0.9.0 · 5879 in / 1503 out tokens · 39992 ms · 2026-05-21T18:25:17.578487+00:00 · methodology

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages · 1 internal anchor

  1. [1]

    A note on Poisson’s equation for pressure in a turbulent flow

    Adrian, Ronald J.1982 Comment on “A note on Poisson’s equation for pressure in a turbulent flow”.The Physics of Fluids25(3), 577–577. Anantharamu, Sreevatsa & Mahesh, Krishnan2020 Analysis of wall-pressure fluctuation sources from direct numerical simulation of turbulent channel flow.J. Fluid Mech.898, A17. Bautista, Juan Carlos Cuevas, Ebadi, Alireza, Wh...

    work page 1982

  2. [2]

    Two-component inner--outer scaling model for the wall-pressure spectrum at high Reynolds number

    Klewicki, J.C.2021 Properties of turbulent channel flow similarity solutions.J. Fluid Mech.915, A39. Klewicki, J. C.2013 A description of turbulent wall-flow vorticity consistent with mean dynamics.J. Fluid Mech.737, 176–204. Klewicki, J. C., Priyadarshana, P. J. A. & Metzger, M. M.2008 Statistical structure of the fluctuating wall pressure and its in-pla...

    work page internal anchor Pith review Pith/arXiv arXiv 2021